Thursday, 25 August 2016

calculus - how to strictly prove $sin x



$$\sin xIn most textbooks, to prove this inequality is based on geometry illustration (draw a circle, compare arc length and chord ), but I think that strict proof should be based on analysis reasoning without geometry illustration. Who can prove it? Thank you very much.






ps:





  1. By differentiation, monotonicity and Taylor formula, all are wrong, because (sinx)=cosx must use lim, and this formula must use \sin x< x. This is vicious circle.


  2. If we use Taylor series of \sin x to define \sin x, strictly prove $\sin x



Answer



We can define \sin x as power series. Applying the knowledge of power series, obtain the derivative of \sin x, and then we will easy prove the inequality. Concluding geometry of \sin x, please refer to this.


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